Step | Hyp | Ref
| Expression |
1 | | eqid 2622 |
. . . . . . 7
⊢
(Base‘𝑊) =
(Base‘𝑊) |
2 | 1 | lindff 20154 |
. . . . . 6
⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊)) |
3 | 2 | ancoms 469 |
. . . . 5
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊)) |
4 | 3 | 3adant3 1081 |
. . . 4
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊)) |
5 | | f1f 6101 |
. . . . 5
⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺:𝐾⟶dom 𝐹) |
6 | 5 | 3ad2ant3 1084 |
. . . 4
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹) |
7 | | fco 6058 |
. . . 4
⊢ ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊)) |
8 | 4, 6, 7 | syl2anc 693 |
. . 3
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊)) |
9 | | ffdm 6062 |
. . . 4
⊢ ((𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊) → ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊) ∧ dom (𝐹 ∘ 𝐺) ⊆ 𝐾)) |
10 | 9 | simpld 475 |
. . 3
⊢ ((𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊) → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊)) |
11 | 8, 10 | syl 17 |
. 2
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊)) |
12 | | simpl2 1065 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊) |
13 | 6 | adantr 481 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺:𝐾⟶dom 𝐹) |
14 | | fdm 6051 |
. . . . . . . . . 10
⊢ ((𝐹 ∘ 𝐺):𝐾⟶(Base‘𝑊) → dom (𝐹 ∘ 𝐺) = 𝐾) |
15 | 8, 14 | syl 17 |
. . . . . . . . 9
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom (𝐹 ∘ 𝐺) = 𝐾) |
16 | 15 | eleq2d 2687 |
. . . . . . . 8
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹 ∘ 𝐺) ↔ 𝑥 ∈ 𝐾)) |
17 | 16 | biimpa 501 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝑥 ∈ 𝐾) |
18 | 13, 17 | ffvelrnd 6360 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → (𝐺‘𝑥) ∈ dom 𝐹) |
19 | 18 | adantrr 753 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → (𝐺‘𝑥) ∈ dom 𝐹) |
20 | | eldifi 3732 |
. . . . . 6
⊢ (𝑘 ∈
((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊))) |
21 | 20 | ad2antll 765 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊))) |
22 | | eldifsni 4320 |
. . . . . 6
⊢ (𝑘 ∈
((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) → 𝑘 ≠
(0g‘(Scalar‘𝑊))) |
23 | 22 | ad2antll 765 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → 𝑘 ≠
(0g‘(Scalar‘𝑊))) |
24 | | eqid 2622 |
. . . . . 6
⊢ (
·𝑠 ‘𝑊) = ( ·𝑠
‘𝑊) |
25 | | eqid 2622 |
. . . . . 6
⊢
(LSpan‘𝑊) =
(LSpan‘𝑊) |
26 | | eqid 2622 |
. . . . . 6
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) |
27 | | eqid 2622 |
. . . . . 6
⊢
(0g‘(Scalar‘𝑊)) =
(0g‘(Scalar‘𝑊)) |
28 | | eqid 2622 |
. . . . . 6
⊢
(Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) |
29 | 24, 25, 26, 27, 28 | lindfind 20155 |
. . . . 5
⊢ (((𝐹 LIndF 𝑊 ∧ (𝐺‘𝑥) ∈ dom 𝐹) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠
(0g‘(Scalar‘𝑊)))) → ¬ (𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))) |
30 | 12, 19, 21, 23, 29 | syl22anc 1327 |
. . . 4
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))) |
31 | | f1fn 6102 |
. . . . . . . . . . 11
⊢ (𝐺:𝐾–1-1→dom 𝐹 → 𝐺 Fn 𝐾) |
32 | 31 | 3ad2ant3 1084 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 Fn 𝐾) |
33 | 32 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → 𝐺 Fn 𝐾) |
34 | | fvco2 6273 |
. . . . . . . . 9
⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥))) |
35 | 33, 17, 34 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝐹 ∘ 𝐺)‘𝑥) = (𝐹‘(𝐺‘𝑥))) |
36 | 35 | oveq2d 6666 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → (𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) = (𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥)))) |
37 | 36 | eleq1d 2686 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ↔ (𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))))) |
38 | | simpl1 1064 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝑊 ∈ LMod) |
39 | | imassrn 5477 |
. . . . . . . . . . 11
⊢ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ ran 𝐹 |
40 | | frn 6053 |
. . . . . . . . . . . 12
⊢ (𝐹:dom 𝐹⟶(Base‘𝑊) → ran 𝐹 ⊆ (Base‘𝑊)) |
41 | 4, 40 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊)) |
42 | 39, 41 | syl5ss 3614 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊)) |
43 | 42 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊)) |
44 | | imaco 5640 |
. . . . . . . . . 10
⊢ ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) |
45 | 15 | difeq1d 3727 |
. . . . . . . . . . . . . . 15
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (dom (𝐹 ∘ 𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥})) |
46 | 45 | imaeq2d 5466 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥}))) |
47 | | df-f1 5893 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐺:𝐾–1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun ◡𝐺)) |
48 | 47 | simprbi 480 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺:𝐾–1-1→dom 𝐹 → Fun ◡𝐺) |
49 | 48 | 3ad2ant3 1084 |
. . . . . . . . . . . . . . 15
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → Fun ◡𝐺) |
50 | | imadif 5973 |
. . . . . . . . . . . . . . 15
⊢ (Fun
◡𝐺 → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥}))) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥}))) |
52 | 46, 51 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥}))) |
53 | 52 | adantr 481 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥}))) |
54 | | fnsnfv 6258 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺 Fn 𝐾 ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥})) |
55 | 32, 54 | sylan 488 |
. . . . . . . . . . . . . 14
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → {(𝐺‘𝑥)} = (𝐺 “ {𝑥})) |
56 | 55 | difeq2d 3728 |
. . . . . . . . . . . . 13
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ {(𝐺‘𝑥)}) = ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥}))) |
57 | | imassrn 5477 |
. . . . . . . . . . . . . . 15
⊢ (𝐺 “ 𝐾) ⊆ ran 𝐺 |
58 | 6 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → 𝐺:𝐾⟶dom 𝐹) |
59 | | frn 6053 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺:𝐾⟶dom 𝐹 → ran 𝐺 ⊆ dom 𝐹) |
60 | 58, 59 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ran 𝐺 ⊆ dom 𝐹) |
61 | 57, 60 | syl5ss 3614 |
. . . . . . . . . . . . . 14
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ 𝐾) ⊆ dom 𝐹) |
62 | 61 | ssdifd 3746 |
. . . . . . . . . . . . 13
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ {(𝐺‘𝑥)}) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)})) |
63 | 56, 62 | eqsstr3d 3640 |
. . . . . . . . . . . 12
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐺 “ 𝐾) ∖ (𝐺 “ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)})) |
64 | 53, 63 | eqsstrd 3639 |
. . . . . . . . . . 11
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)})) |
65 | | imass2 5501 |
. . . . . . . . . . 11
⊢ ((𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺‘𝑥)}) → (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))) |
66 | 64, 65 | syl 17 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → (𝐹 “ (𝐺 “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))) |
67 | 44, 66 | syl5eqss 3649 |
. . . . . . . . 9
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))) |
68 | 1, 25 | lspss 18984 |
. . . . . . . . 9
⊢ ((𝑊 ∈ LMod ∧ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})) ⊆ (Base‘𝑊) ∧ ((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))) → ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))) |
69 | 38, 43, 67, 68 | syl3anc 1326 |
. . . . . . . 8
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ 𝐾) → ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))) |
70 | 17, 69 | syldan 487 |
. . . . . . 7
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)})))) |
71 | 70 | sseld 3602 |
. . . . . 6
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) → (𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))) |
72 | 37, 71 | sylbid 230 |
. . . . 5
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹 ∘ 𝐺)) → ((𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) → (𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))) |
73 | 72 | adantrr 753 |
. . . 4
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ((𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥}))) → (𝑘( ·𝑠
‘𝑊)(𝐹‘(𝐺‘𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺‘𝑥)}))))) |
74 | 30, 73 | mtod 189 |
. . 3
⊢ (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹 ∘ 𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))) |
75 | 74 | ralrimivva 2971 |
. 2
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))) |
76 | | simp1 1061 |
. . 3
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝑊 ∈ LMod) |
77 | | rellindf 20147 |
. . . . . 6
⊢ Rel
LIndF |
78 | 77 | brrelexi 5158 |
. . . . 5
⊢ (𝐹 LIndF 𝑊 → 𝐹 ∈ V) |
79 | 78 | 3ad2ant2 1083 |
. . . 4
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐹 ∈ V) |
80 | | simp3 1063 |
. . . . . 6
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺:𝐾–1-1→dom 𝐹) |
81 | | dmexg 7097 |
. . . . . . 7
⊢ (𝐹 ∈ V → dom 𝐹 ∈ V) |
82 | 79, 81 | syl 17 |
. . . . . 6
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → dom 𝐹 ∈ V) |
83 | | f1dmex 7136 |
. . . . . 6
⊢ ((𝐺:𝐾–1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V) |
84 | 80, 82, 83 | syl2anc 693 |
. . . . 5
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐾 ∈ V) |
85 | | fex 6490 |
. . . . 5
⊢ ((𝐺:𝐾⟶dom 𝐹 ∧ 𝐾 ∈ V) → 𝐺 ∈ V) |
86 | 6, 84, 85 | syl2anc 693 |
. . . 4
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → 𝐺 ∈ V) |
87 | | coexg 7117 |
. . . 4
⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V) |
88 | 79, 86, 87 | syl2anc 693 |
. . 3
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) ∈ V) |
89 | 1, 24, 25, 26, 28, 27 | islindf 20151 |
. . 3
⊢ ((𝑊 ∈ LMod ∧ (𝐹 ∘ 𝐺) ∈ V) → ((𝐹 ∘ 𝐺) LIndF 𝑊 ↔ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))))) |
90 | 76, 88, 89 | syl2anc 693 |
. 2
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → ((𝐹 ∘ 𝐺) LIndF 𝑊 ↔ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom (𝐹 ∘ 𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖
{(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠
‘𝑊)((𝐹 ∘ 𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹 ∘ 𝐺) “ (dom (𝐹 ∘ 𝐺) ∖ {𝑥})))))) |
91 | 11, 75, 90 | mpbir2and 957 |
1
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊) |